Existence Results for a Class of Periodic Evolution Variational Inequalities*

In this paper, using the Brouwer topological degree, we prove an existence result for finite variational inequalities. This approach is also applied to obtain the existence of periodic solutions for a class of evolution variational inequalities.

Sufficient conditions for stability on a finite time interval are derived for a class of evolution variational inequalities with the use of Lyapunov functions and frequency-domain conditions. These inequalities are considered for Hilbert spaces and Sobolev spaces of infinite order in the sense of Yu.A. Dubinskii. We show how to use the stability result on a finite interval to characterize bifurcations. An algorithm for finding observation functionals is presented that uses an isomorphism between the algebra of pseudodifferential operators with constant coefficients whose symbols are real-analytic functions in some domain and an algebra of analytic matrix-valued functions.

This paper gives new existence results for elliptic and evolutionary variational and quasi-variational inequalities. Specifically, we give an existence theorem for evolutionary variational inequalities involving different types of pseudo-monotone operators. Another existence result embarks on elliptic variational inequalities driven by maximal monotone operators. We propose a new recessivity assumption that extends all the classical coercivity conditions. We also obtain criteria for solvability of general quasi-variational inequalities treating in a unifying way elliptic and evolutionary problems. Two of the given existence results for evolutionary quasi-variational inequalities rely on Mosco-type continuity properties and Kluge's fixed point theorem for set-valued maps. We also focus on the case of compact constraints in the evolutionary quasi-variational inequalities. Here a relevant feature is that the underlying space is the domain of a linear, maximal monotone operator endowed with the graph norm. Applications are also given.

Preface. Introduction. 1. Elements of Nonsmooth Analysis. 1. Generalized Gradients of Locally Lipschitz Functionals. 2. Palais Smale Condition and Coerciveness for a Class of Nonsmooth Functionals. 3. Nonsmooth Analysis in the Sense of Degiovanni. 2. Variational Methods. 1. Critical Point Theory for Locally Lipschitz Functionals. 2. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals. 3. A Critical Point Theory in Metric Spaces. 3. Variational Methods. 1. Critical Point Theory for Convex Perturbations of Locally Lipschitz Functionals in the Limit Case. 2. Examples. 4. Multivalued Elliptic Problems in Variational Form. 1. Multiplicity for Locally Lipschitz Periodic Functionals. 2. The Multivalued Forced-pendulum Problem. 3. Hemivariational Inequalities Associated to Multivalued Problems with Strong Resonance. 4. A Parallel Nonsmooth Critical Point Theory. Approach to Stationary Schrodinger Type Equations in Constraints. 8. Non-Symmetric Perturbations of Symmetric Eigenvalue Problems. 1. Non-Symmetric Perturbations of Eigenvalue Problems for Periodic Hemivariational Inequalities with Constraints. 2. Perturbations of Double of Eigenvalue Problems for General Hemivariational Inequalities with Constraints. 3. location of Solutions by Minimax Methods of Variational Hemivariational Inequalities. 10. Nonsmooth Evolution Problems. 1. First Order Evolution Variational Inequalities. 2. Second Order Evolution Variational Inequalities. 3. Stability Problems for Evolution Variational Inequalities. 11. Inequality Problems in BV and Geometric Applications. 1. The General Framework. 2. Area Type Functionals. 3. A Result of Clark Type. 4. An Inequality Problem with Superlinear Potential.

In this paper, we study a class of subdifferential evolution inclusions involving history-dependent operators. First, we improve an existence and uniqueness theorem and prove the continuous dependence result in the weak topologies. Next, we establish the existence of optimal solution to an optimal control problem for the evolution inclusion. Finally, we illustrate the results by an example of an optimal control of a dynamic frictional contact problem in mechanics, whose weak formulation is the evolution variational inequality.

The aim of this paper is to obtain the continuity of solutions to time-dependent nonlinear variational and quasi-variational inequalities which express many dynamic equilibrium problems. To prove our results, we make use of Minty's Lemma and of the notion of the Mosco's convergence.

The aim of this paper is to study the Walrasian equilibrium problem when the data are time-dependent. In order to have a more realistic model, the excess demand function depends on the current price and on previous events of the market. Hence, a memory term is introduced; it describes the precedent states of the equilibrium. This model is reformulated as an evolutionary variational inequality in the Lebesgue space L2([0,T],ℝ), and, thanks to this characterization, existence and qualitative results on equilibrium solution are given.

In this paper, we develop a cooperative game framework for modeling the pollution control problem in a time-dependent setting. We examine the situation in which different countries, aiming at reducing pollution emissions, coordinate both emissions and investment strategies to optimize jointly their welfare. We state the equilibrium conditions underlying the model and provide a formulation in terms of an evolutionary variational inequality. Then, by means of infinite dimensional duality tools, we prove the existence of Lagrange multipliers that play a fundamental role to describe countries’ decision-making processes. Finally, we discuss the existence of solutions and provide a numerical example.

Variational formulation for a general dynamic financial equilibrium problem: Balance law and liability formula

This book offers a comprehensive analysis of dynamic networks and evolutionary variational inequalities, a topic of growing prominence in the study of networks. The extraordinary importance of networks in finance, mathematics, computer science and other areas is well known but the relatively new concept of ‘dynamic’ networks is less well understood. They become dynamic when the constitutive elements of the phenomena associated with the fixed geometry of networks are considered to be evolving over time. Patrizia Daniele offers many numerical examples to illustrate the issues discussed and provides a broad appendix to enrich this challenging but deeply informative book.

Variational principles of non-linear theory of brittle fracture

A λ-convexity based proof for the propagation of chaos for weakly interacting stochastic particles

Contents: 1. Introduction 2. The Traffic Equilibrium Problem 3. Evolutionary Spatial Price Equilibrium 4. The Evolutionary Financial Model 5. Projected Dynamical Systems A. Definitions and Properties B. Weak Convergence C. Generalized Derivatives D. Variational Inequalities E. Quasi-Variational Inequalities F. Infinite Dimensional Duality Bibliography Index

Evolution variational inequalities with general costs

Some Topics on Variational Inequalities and Applications